All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$4.00$ each for students, and the group paid $$57.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$17.00$ each for teachers and $$10.50$ each for students, and the group paid $$139.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4y = 57}$ ${17x+10.5y = 139}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-8y = -114}$ ${17x+10.5y = 139}$ Add the top and bottom equations together. $ 2.5y = 25 $ $ y = \dfrac{25}{2.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+4y = 57}$ to find $x$ ${8.5x + 4}{(10)}{= 57}$ $8.5x+40 = 57$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {17x+10.5y = 139}$ and get the same answer for $x$ ${17x + 10.5}{(10)}{= 139}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.